# Modeling, Functions, and Graphs

## Section9.3Series

### SubsectionIntroduction

#### Checkpoint9.47.Practice 3.

Arlene from the previous example decides to start a savings account. After her first paycheck, she deposits $50, and plans to increase her deposits by$5 per month. How much will she have in the savings account at the end of the 18-month training program?
$Answer. $$1665$$ Solution.$1665

### SubsectionGeometric Series

We can also find a formula for the sum of the first $$n$$ terms of a geometric sequence.

#### Investigation9.3.Sum of a Geometric Sequence.

We’ll compute the sum of the first 9 terms of the geometric sequence with $$a=3$$ and $$r=3\text{.}$$ Its general term is $$a_n = 3(3)^{n-1}\text{,}$$ or $$3^n\text{.}$$ Then
\begin{equation*} S_9 = 3 + 3^2 + 3^3 + \cdots + 3^9 \end{equation*}
1. Instead of writing the terms in the opposite order, as we did for the arithmetic series, this time we will multiply each term of $$S_9$$ by the common ratio, 3.
\begin{equation*} 3 \cdot S_9 = 3 \cdot 3 + 3 \cdot 3^2 + 3 \cdot 3^3 + \cdots + 3 \cdot 3^8 + 3 \cdot 3^9 \end{equation*}
or
\begin{equation*} 3S_9 = 3^2 + 3^3 + 3^4 + \cdots + 3^9 + 3^{10} \end{equation*}
2. Next we subtract the first equation from the second equation. Most of the terms will cancel out in the subtraction.
\begin{equation*} \begin{aligned}[t] 3S_9 \amp = ~~~~~~~~~~~3^2 + 3^3 + \cdots + 3^9 + 3^{10}\\ -[S_9 \amp = ~~~3 + 3^2 + 3^3 + \cdots + 3^9]\\ 2S_9 \amp = -3 ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + 3^{10}\\ \end{aligned} \end{equation*}
3. Thus
\begin{equation*} 2S_9 = 3^{10} - 3 \end{equation*}
so
\begin{equation*} S_9 = \dfrac{(\hphantom{0000}) - (\hphantom{0000})}{(\hphantom{0000})} = \end{equation*}
4. Notice that the second term of the numerator is $$a_1$$ (3 in this example) and the first term is $$a_{10}~ (3 \cdot 3^9$$ or $$3^{10})\text{.}$$
5. Although you might not guess it from this example, the denominator of the expression, 2, is equal to $$r-1\text{.}$$ In general, we can derive the following formula for computing a geometric series.

#### Geometric Series.

The sum $$S_n$$ of the first $$n$$ terms of a geometric sequence is
\begin{equation*} S_n = \dfrac{a_{n+1} - a_1}{r-1} \end{equation*}

#### Checkpoint9.48.QuickCheck 2.

To find the series for a given sequence, we
• Compute the last term of the sequence.
• Find the average value of the terms.
• List the terms in the opposite order.
$$\text{Add the terms.}$$
Solution.

#### Example9.49.

Find the sum of the first five terms of the sequence $$5, \dfrac{10}{3}, \dfrac{20}{9}, \dfrac{40}{27}, \cdots\text{.}$$
Solution.
The sequence is geometric with $$a_1 = 5\text{.}$$ We find by dividing $$\dfrac{10}{3}$$ by 5.
\begin{equation*} \dfrac{10}{3} \div 5 = \dfrac{2}{3} \end{equation*}
Thus, the general term of the series is $$a_n = 5\left(\dfrac{2}{3}\right)^{n-1}\text{.}$$ We use the formula for a geometric series with $$n=5\text{,}$$
\begin{equation*} S_5 = \dfrac{a_6 - a_1}{r-1} \end{equation*}
We substitute $$a_1 = 5,~a_6 = 5\left(\dfrac{2}{3}\right)^5\text{,}$$ and $$r=\dfrac{2}{3}\text{.}$$ Thus
\begin{equation*} \begin{aligned}[t] S_5 \amp = \dfrac{5\left(\dfrac{2}{3}\right)^5 - 5}{\dfrac{2}{3}-1} = \dfrac{5\left(\dfrac{32}{243} - 1\right)}{-\dfrac{1}{3}}\\ \amp = 5\left(\dfrac{32-243}{243}\right)\left(\dfrac{-3}{1}\right) \approx 13.02\\ \end{aligned} \end{equation*}

#### Checkpoint9.50.Practice 4.

Find the sum of the first ten terms of the geometric sequence $$100\text{,}$$ $$110\text{,}$$ $$121,~ \ldots\text{.}$$
$$1593.74$$
Solution.
1593.74

#### Example9.51.

Payam’s starting salary as an engineer is 60,000 with a 5% annual raise for each of the next five years, depending on suitable progress. If Payam receives each salary increase, how much will he make over the next six years? Solution. Payam’s salary is multiplied each year by a factor of 1.05, so its values form a geometric sequence with a common ratio of $$r=1.05\text{.}$$ The general term of the sequence is $$a_n = 60,000(1.05)^{n-1}\text{.}$$ His total income over the six years will be the sum of the first six terms of the sequence. Thus, \begin{equation*} \begin{aligned}[t] S_6 \amp = \dfrac{a_7 - a_1}{r-1} = \dfrac{60,000(1.05)^6 - 60,000}{1.05-1}\\ \amp = 408,114.77 \\ \end{aligned} \end{equation*} Payam will earn408,114.77 over the next six years.

#### Checkpoint9.52.Practice 5.

Show that the formula for a geometric series can also be written as
\begin{equation*} \begin{gathered} S_n = \dfrac{a_1(r^n - 1)}{r-1} \end{gathered} \end{equation*}
Check all the relevant statements.
1. The sum $$S_n$$ of the first $$n$$ terms of a geometric series is
• $$\displaystyle S_n = \dfrac{a_{n+1} - a_1}{r-1}$$
• $$\displaystyle S_n = \dfrac{n}{2}(a_{1} + a_n)$$
• $$\displaystyle S_n = a_1 r^n$$
• $$\displaystyle S_n = a + (n-1)r$$
2. We can write $$a_{n+1}$$ as
• $$\displaystyle a_{n+1} = a_{1} + nr$$
• $$\displaystyle a_{n+1} = a_{1} + (n-1)r$$
• $$\displaystyle a_{n+1} = a_1 r^n$$
• $$\displaystyle a_{n+1} = a_1 r^{n-1}$$
3. So the numerator from part (a) can be rewritten as
• $$\displaystyle a_{1} + nr -a_1 = a_{1} (r^n-1)$$
• $$\displaystyle a_{1} + (n-1)r - a_1 = a_{1} (r^n-1)$$
• $$\displaystyle a_1 r^n -a_1 = a_{1} (r^n-1)$$
• $$\displaystyle a_1 r^{n-1} - a_1 = a_{1} (r^n-1)$$
$$\text{Choice 1}$$
$$\text{Choice 3}$$
$$\text{Choice 3}$$
Solution.
$$S_n = \dfrac{a_{n+1} - a_1}{r-1} = \dfrac{a_1r^n - a_1}{r-1} = \dfrac{a_1(r^n - 1)}{r-1}$$

### SubsectionSection Summary

#### SubsubsectionVocabulary

• Series
• Arithmetic series
• Geometric series

#### SubsubsectionCONCEPTS

1. The sum of the terms of a sequence is called a series.
2. The sum $$S_n$$ of the first $$n$$ terms of an arithmetic sequence is
\begin{equation*} S_n = \dfrac{n}{2} (a_1 + a_n) \end{equation*}
3. The sum $$S_n$$ of the first $$n$$ terms of a geometric sequence is
\begin{equation*} S_n = \dfrac{a_{n+1} - a_1}{r-1} \end{equation*}

#### SubsubsectionSTUDY QUESTIONS

1. What is the difference between a sequence and a series?
2. Describe a method for finding the sum of an arithmetic sequence.
3. Describe a method for finding the sum of a geometric sequence.
4. How can you decide whether a given sequence is arithmetic or geometric (or neither)?

#### SubsubsectionSKILLS

Practice each skill in the Homework Problems listed.
1. Evaluate arithmetic and geometric series: #1-12
2. Identify a series as arithmetic or geometric: #13-22
3. Model a situation with a series: #23-40

### ExercisesHomework 9.3

#### Exercise Group.

For Problems 1–6, evaluate the arithmetic series.
##### 1.
The sum of the first nine terms of the sequence $$~a_n = -4 + 3n$$
##### 2.
The sum of the first ten terms of the sequence $$~a_n = 5-2n$$
##### 3.
The sum of the first 16 terms of the sequence $$~a_n = 18-\dfrac{4}{3}n$$
##### 4.
The sum of the first 13 terms of the sequence $$~a_n = -6 - \dfrac{1}{2}n$$
##### 5.
The sum of the first 30 terms of the sequence $$~a_n = 1.6+0.2n$$
##### 6.
The sum of the first 25 terms of the sequence $$~a_n = 2.5+0.3n$$

#### Exercise Group.

For Problems 7–12, evaluate the geometric series.
##### 7.
The sum of the first five terms of $$a_n = 2(-4)^{n-1}$$
##### 8.
The sum of the first eight terms of $$a_n = 12(3)^{n-1}$$
##### 9.
The sum of the first nine terms of $$a_n = -48\left(\dfrac{1}{2}\right)^{n-1}$$
##### 10.
The sum of the first six terms of $$a_n = 81\left(\dfrac{2}{3}\right)^{n-1}$$
##### 11.
The sum of the first four terms of $$a_n = 18(1.15)^{n-1}$$
##### 12.
The sum of the first four terms of $$a_n = 512(0.72)^{n-1}$$

#### Exercise Group.

For Problems 13–22, identify the series as arithmetic, geometric or neither, then evaluate it.
##### 13.
$$2+4+6+\cdots+96+98+100$$
##### 14.
$$1+3+5+\cdots+95+97+99$$
##### 15.
$$2+4+8+16+\cdots+256+512+1024$$
##### 16.
$$1+3+9+27+\cdots+6561+19,683$$
##### 17.
$$1+8+24+64+125+216+343$$
##### 18.
$$1+11+111+1111+11,111+111,111$$
##### 19.
$$87+84+81+78+\cdots+45+42+39$$
##### 20.
$$1+(-2)+(-5)+\cdots+(-41)+(-44)$$
##### 21.
$$6+2+\dfrac{2}{3}+\cdots+\dfrac{2}{81}+\dfrac{2}{243}$$
##### 22.
$$12+3+\dfrac{3}{4}+\cdots+\dfrac{3}{64}+\dfrac{3}{128}$$

#### Exercise Group.

For Problems 23–40, write a series to describe the problem, then evaluate it.
##### 23.
Find the sum of all the even integers from 14 to 88.
##### 24.
Find the sum of all multiples of 7 from 14 to 105.
##### 25.
A clock strikes once at one o’clock, twice at two o’clock, and so on. How many times will the clock strike in a twelve hour period?
##### 26.
Jessica puts one candle on the cake at her daughter’s first birthday, two candles at her second birthday, and so on. How many candles will Jessica have used after her daughter’s sixteenth birthday?
##### 27.
A rubber ball is dropped from a height of 24 feet and returns to three-fourths of its previous height on each bounce.
1. How high does the ball bounce after hitting the floor for the third time?
2. How far has the ball traveled vertically when it hits the floor for the fourth time?
##### 28.
A Yorkshire terrier can jump 3 feet into the air on his first bounce and five-sixths the height of his previous jump on each successive bounce.
1. How high can the terrier go on his fourth bounce?
2. How far has the terrier traveled vertically when he returns to the ground after his fourth bounce?
##### 29.
Sales of Brussels Sprouts dolls peaked at $920,000 in 1991 and began to decline at a steady rate of$40,000 per year. What total revenue did the manufacturer gain from sale of the dolls from 1991 to 2000?
##### 30.
It takes Alida 20 minutes to type the first page of her term paper, but each subsequent page takes her 40 seconds less than the previous one. How long will it take her to type her 30-page paper?
##### 31.
A computer takes 0.1 second to perform the first iteration of a certain loop, and each subsequent iteration takes 0.05 seconds longer than the previous one. How long will it take the computer to perform 50 iterations?
##### 32.
Richard’s water bill was $63.50 last month. If his bill increases by$2.30 per month, how much should he expect to pay for water during the next 10 months?
##### 39.
Suppose that you are given 1¢ on the first day of the month, 2¢ on the second day, 4¢ on the third day, and so on, each day’s payment being twice the previous day’s. What would be your total income on the thirtieth day?
##### 40.
According to legend, a man who had pleased the Persian king asked for the following reward. The man was to receive a single grain of wheat for the first square of a chessboard, two grains for the second square, four grains for the third square, and so on, doubling the amount for each square up to the sixty-fourth square. How many grains would he receive in all?